My Higgs story

M. Veltman

Paris, 30 july 2015

NIKHEF

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Higgs particle…. What does it mean ?What does it do ? It is claimed to give massto all other particles. What does that mean ?Can we now predict the masses of all particles ?

In this talk it will be attempted to explain why the Higgs particleis important to the theory of the Standard Model.

No, it predicts nothing except its own existence.But it does not even predict its own mass.

But other facts are very important as well. For example, theremust be a build-in symmetry. Of course, in the end, the real reasonfor the importance of a theory is that one can calculate processes,and check if they agree with experiment.

The self-energy of the electron.

In that theory the electron is assumed to be a small sphere withradius r. The charge is smeared out over this sphere. The self-energy is the energy of this sphere assuming that it is zero ifr is infinite:

It is inversely proportional to the radius r.

The smearing out of the charge is called a regulator mechanism.The radius r is the regulator parameter. If r goes to zero the energyE becomes infinite inversely proportional to this radius. Onespeaks of a linear divergence.This description remained part of the theory for some 35 years.

1904: theory of Abraham and Lorentz.

Standard quantummechanics cannot describe processes involvingthe creation of new particles. To treat the latter type of processesone needs quantum field theory.

Quantum field theory was created in 1929 by Heisenberg andPauli. The calculation of the electron self-energy was doneby Weisskopf in 1939.

The result was still infinite, except that now the degree ofdivergence was logarithmic. Here is how that goes in simplifiedform:

Self-energy Feynman diagram Corresponding expression:

That expression is linearly divergent. However that becomeslogarithmic if we use symmetric integration:

We see here the following. If you want to deal with infiniteobjects you have to invent a way to make them finite, thatis to invent a regulator mechanism. And then, given aregulator parameter, take the limit to reality.

There is another important property, which is the conservationof electric charge. That is a consequence of gauge invarianceof the theory. The regulator method should respect that property.

Pauli and Villars understood this. They developed a regulatormethod for quantum electrodynamics that respected gaugeinvariance and Lorentz invariance.

A new complication arises: the regulator mechanism mayviolate important properties. For example, in the abovecase Lorentz invariance is violated. A sphere will not remaina sphere when subjected to a Lorentz transformation. Thismay result in unwanted effects.

Then there was the great revolution of 1948.

There were two experiments that gave results that could not beexplained by ordinary quantum mechanics: the Lamb shift andthe anomalous magnetic moment of the electron. Everyone feltthat quantum field theory was needed here, but there was reallyno way of doing that and every calculation got stuck in infinities.

Then Kramers (Leiden) came with his idea of renormalization.

Consider the electron self-energy. There is, in addition to the‘bare mass’, the energy due to the electric field of the electronitself (which is infinite). De sum of the two is the observedmass of the electron.

A theory where all infinities can be absorbed in the available freeparameters is called a renormalizable theory. After absorbing theinfinities observable quantities can be computed.

But, argued Kramers, no one knows the mass of un unchargedelectron, so why not absorb the (infinite) self-energy in this(unknown) bare mass.

The idea of Kramers made all the difference. Calculationswere done by Bethe, Feynman, Schwinger and Tomonaga,giving results that agreed with experiment. The resultingtheory is the renormalizable theory of quantumelectrodynamics.However, the most important result was that of Feynman. Hedeveloped a simple method to do the actual calculations.For any given calculation make a few drawings (called Feynmandiagrams), and subsequently using simple rules (the Feynman rules)one can write down the expressions to be calculated.

After receiving the Nobel prize in 1965 Feynman gave alecture at CERN and he asked the rhetorical question: whatdid I really do ? I just constructed a bookkeeping scheme. But in a complicated theory (such as field theory) simplificationis very necessary and the key to further progress.

In other words: given the Feynman rules for a given theoryyou can compute every experimental prediction of that theory. Dyson developed the formal theory, deriving these results withinthe canonical formalism, thus guaranteeing unitarity.

Feynman’s car.

1959. Scottish summer school.

At this school Chew preached the endof quantum field theory.

Instead he proposed S-matrix theory,based on ideas involving analyticalproperties of scattering amplitudes.

Some participants: Cabibbo, Robinson, Glashow, Veltman.Higgs had the key of the wine cellar. He described de group aboveas the gang of four, as we kept pestering him about the wine.

We did not believe anything from that.

Personally I thought that unstableparticles made for a counterexample.

In the end, looking back, all of these statements are wrong.

Unstable particlesSo I started working on unstable particles. The funny thing was thatthe Feynman rules were more or less generally known, but nobodyhad developed a consistent theory. There existed no clear derivationof these rules, although the result seemed correct. What to do ?

Like with many ‘problems’ that I have encountered, the answeris that there is no problem. Just think about it clearly.

For any given theory the procedure is simply. Start deriving theFeynman rules. The derivation goes in some standard way (Dyson)That guarantees that a number of fundamental properties hold. Butit happens that for unstable particles that approach breaks down.We know the result but do not know how to derive it.

Well, it is simple. Check if these fundamental properties hold.Then there is no need to worry where these rules came from.

What are these fundamental properties ? They are unitarity(conservation of probability) and causality.

So I started working on the problem of showing that a givenset of Feynman rules is correct, and in about 1 year I founda way to solve that problem.

As you can see the publication date was three years later for nogood reason. Also, Physica was mainly used for publications onstatistical mechanics, not particle physics. For all I know thearticle was for at least 5 years read by only one person, Symanzik.

In any case, after this I worked with Feynman rules of which Icould see if they were correct, and did not care where they didcome from.

In 1961 I moved to CERN, following my thesis advisor Leon van Hove,staying there till 1966, learning a lot about particle physics experiments.I more or less joined the big (for that time) CERN neutrino experimentand in fact I was their spokesman at the Brookhaven conference in1963 (140 participants, of which 15 had or would get the Nobelprize).

Spark chamber Bubble chamber

van der Meer’s magnetic horn Simon van der Meer

To make a long story short, I returned to Utrecht in 1966; in 1968I had convinced myself that the weak interactions were some formof a Yang-Mills theory. I could not have done that without theknowledge of experimental physics that I had acquired at CERN.Weak interactions. A collective for a large number of phenomenaobserved experimentally and characterized by their relativeweakness. Examples: neutron decay, muon decay, neutrino events.The interactions had a well-known structure, at least in lowestorder. This structure strongly suggested the existence of spin 1particles, called vector bosons. Examples:

Neutron decay.

K-meson decay:

No one knew how to attack these interactions beyond the lowestorder. It was the main problem of those days.

A Yang-Mills theory is a theory about vector particles interactingwith each other in precisely described manner (there is asymmetry). There is a three-point and a four-point interaction:

So I started to investigate this theory of mutually interactingvector bosons, with the hope that it would be a renormalizabletheory. The starting point was terrible: the Feynman rules forthis theory let to very, very bad divergencies.For example I looked at WW scattering andfound divergencies of the form infinity to theeight power at the one loop level.

Salam claimed to have provennon-renormalizability.

This was the state of affairs in 1968.

Then I discovered an important thing. First of all, there were manymore diagrams than in the usual renormalizable theories, andsecondly, there were many cancellations between the diagrams,due to the symmetry of the theory.The question was how to attack this situation. It was quite hopelessto work out the diagrams and check that the infinities cancel.There were just too many of them, all these divergencies led tounclear and murky situations. Even with a computer (actually,in the period 1966-1972 no suitable computer was at my disposalin Utrecht) it was quite impossible to do things. Besides, computershave no ideas. It was a total mess.There was only one thing that I could think of. I must try toinvent alternative Feynman rules such that:-  physics would be unchanged;-  all cancellations were somehow already part of the new rules.I found a way to derive such new rules, involving ghost particles,and miraculously almost all infinities disappeared. This was 1968.

To get an idea: here some contributingdiagrams to WW scattering.

De way the theory looked like at that time was pretty terrible. Butgradually I cleaned up piece by piece. After a while it became clearthat not all was fine. The theory was not quite renormalizable.The trouble was with the masses of the vector bosons. The masslesscase seemed renormalizable. In a pure Yang-Mills theory the masses of the vector bosons arezero. On the other hand, from the experiment, it was clear thatthe vector bosons of weak interactions were quite massive. I did not know how to attack thissituation. Enter the sigma model (ofSchwinger) in the form of Hugo Strubbe.Strubbe was a Belgian who wantedto come to the Institute. He hadworked on the sigma model ofSchwinger, and wanted tocontinue his work in Utrecht.

Hugo Strubbe, in UtrechtOct. 1970 – Dec. 1971

The sigma model (Schwinger 1957) was a theory of a system of threepions and one further spinless particle called the sigma. This sigmaparticle had interactions with the pions and also self-interactions suchthat there was a state with everywhere a sigma field present but with anenergy less than zero, that is less than the vacuum.Therefore the vacuum decays into the sigma vacuum which thenbecomes the standard vacuum.Thus in this model the sigma vacuum replaces the usual one.If now some particle has an interaction with that sigma fieldin the vacuum it will get some potential energy, manifest asa mass. In other words, here is a way to give particles a mass.Choosing the strength of the coupling to the sigma field onecan get any mass.

Hugo Strubbe gave a lecture about that when arriving at theInstitute, and ‘t Hooft attending that seminar got the idea toapply this to generate a mass for the vector bosons.

Schwinger used this method to give a mass to the muon.

And here was again a miracle: the last difficulties in the Yang-Millstheory disappeared and the theory became fully renormalizable,that is all occurring infinities could be absorbed in the availablefree parameters. Theories with a Yang-Mills structure were now renormalizable theories.

‘t Hooft presented this result at the 1971 conference in Amsterdamwhich created a revolution.

The next issue was to find the precise model for the weak interactions.Such a model existed already although it had received virtually noattention. It was proposed by Weinberg in 1967 but actually nevermentioned by him before 1972.

Now what about Higgs ? In connection with ‘t Hooft’s paper itwas discovered that giving vector bosons a mass using a sigmavacuum was already treated by Higgs and others. They hadintroduced a model in connection with superconductivity.

What happened is this. It was discovered that within a superconductorelectromagnetic fields have a finite range. In particle physics thatis what you get if the particle has a mass. So Higgs (and Englertand Brout) started to work on the problem of how to give a photona mass. More specifically, they wanted the photon to have a massinside a superconductor but not outside. The solution was to assumesome field inside the superconductor that would, like in the sigmamodel, give a mass to a particle moving in that field.

Higgs discovered that unavoidably such a construction wouldgive rise to another particle, what we now call the Higgs particle.

Also in the cases that ‘t Hooft discussed there was such a Higgsparticle, except he did not call it that. Later that was corrected

Higgs: 1929

Conclusion

The importance of the Higgs construction is that it made the theoryof Yang-Mills fields renormalizable. Observable results can becalculated and compared with experiment, and that has happenedin a multitude of ways in the last 40 years, up to and including therecent discovery of the Higgs particle.

At this point one may ask:-  why does nature at low energies use only renormalizable theories ?-  what about gravitation ?

There are other questions:-  why three generations of quarks and leptons ?-  why masses ranging from eV to 180 GeV ?-  etc.

Have fun…